3.2.15 \(\int e^{-x} (-144 x^3+36 x^4+e^x (15+24 x^2-6 x \log (7))) \, dx\) [115]

Optimal. Leaf size=27 \[ x \left (4 x^2 \left (2-9 e^{-x} x\right )+3 (5-x \log (7))\right ) \]

[Out]

x*(4*(2-9*x/exp(x))*x^2+15-3*x*ln(7))

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Rubi [A]
time = 0.15, antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 12, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6874, 2207, 2225} \begin {gather*} -36 e^{-x} x^4+8 x^3-3 x^2 \log (7)+15 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-144*x^3 + 36*x^4 + E^x*(15 + 24*x^2 - 6*x*Log[7]))/E^x,x]

[Out]

15*x + 8*x^3 - (36*x^4)/E^x - 3*x^2*Log[7]

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-144 e^{-x} x^3+36 e^{-x} x^4+3 \left (5+8 x^2-2 x \log (7)\right )\right ) \, dx\\ &=3 \int \left (5+8 x^2-2 x \log (7)\right ) \, dx+36 \int e^{-x} x^4 \, dx-144 \int e^{-x} x^3 \, dx\\ &=15 x+8 x^3+144 e^{-x} x^3-36 e^{-x} x^4-3 x^2 \log (7)+144 \int e^{-x} x^3 \, dx-432 \int e^{-x} x^2 \, dx\\ &=15 x+432 e^{-x} x^2+8 x^3-36 e^{-x} x^4-3 x^2 \log (7)+432 \int e^{-x} x^2 \, dx-864 \int e^{-x} x \, dx\\ &=15 x+864 e^{-x} x+8 x^3-36 e^{-x} x^4-3 x^2 \log (7)-864 \int e^{-x} \, dx+864 \int e^{-x} x \, dx\\ &=864 e^{-x}+15 x+8 x^3-36 e^{-x} x^4-3 x^2 \log (7)+864 \int e^{-x} \, dx\\ &=15 x+8 x^3-36 e^{-x} x^4-3 x^2 \log (7)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.08, size = 30, normalized size = 1.11 \begin {gather*} 3 \left (5 x+\frac {8 x^3}{3}-12 e^{-x} x^4-x^2 \log (7)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-144*x^3 + 36*x^4 + E^x*(15 + 24*x^2 - 6*x*Log[7]))/E^x,x]

[Out]

3*(5*x + (8*x^3)/3 - (12*x^4)/E^x - x^2*Log[7])

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Maple [A]
time = 0.03, size = 26, normalized size = 0.96

method	result	size
default	\(8 x^{3}+15 x -3 x^{2} \ln \left (7\right )-36 x^{4} {\mathrm e}^{-x}\)	\(26\)
risch	\(8 x^{3}+15 x -3 x^{2} \ln \left (7\right )-36 x^{4} {\mathrm e}^{-x}\)	\(26\)
norman	\(\left (-36 x^{4}+15 \,{\mathrm e}^{x} x +8 \,{\mathrm e}^{x} x^{3}-3 x^{2} \ln \left (7\right ) {\mathrm e}^{x}\right ) {\mathrm e}^{-x}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x*ln(7)+24*x^2+15)*exp(x)+36*x^4-144*x^3)/exp(x),x,method=_RETURNVERBOSE)

[Out]

8*x^3+15*x-3*x^2*ln(7)-36*x^4/exp(x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (24) = 48\).
time = 0.28, size = 59, normalized size = 2.19 \begin {gather*} 8 \, x^{3} - 3 \, x^{2} \log \left (7\right ) - 36 \, {\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} + 144 \, {\left (x^{3} + 3 \, x^{2} + 6 \, x + 6\right )} e^{\left (-x\right )} + 15 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x*log(7)+24*x^2+15)*exp(x)+36*x^4-144*x^3)/exp(x),x, algorithm="maxima")

[Out]

8*x^3 - 3*x^2*log(7) - 36*(x^4 + 4*x^3 + 12*x^2 + 24*x + 24)*e^(-x) + 144*(x^3 + 3*x^2 + 6*x + 6)*e^(-x) + 15*
x

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Fricas [A]
time = 0.30, size = 32, normalized size = 1.19 \begin {gather*} -{\left (36 \, x^{4} - {\left (8 \, x^{3} - 3 \, x^{2} \log \left (7\right ) + 15 \, x\right )} e^{x}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x*log(7)+24*x^2+15)*exp(x)+36*x^4-144*x^3)/exp(x),x, algorithm="fricas")

[Out]

-(36*x^4 - (8*x^3 - 3*x^2*log(7) + 15*x)*e^x)*e^(-x)

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Sympy [A]
time = 0.04, size = 24, normalized size = 0.89 \begin {gather*} - 36 x^{4} e^{- x} + 8 x^{3} - 3 x^{2} \log {\left (7 \right )} + 15 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x*ln(7)+24*x**2+15)*exp(x)+36*x**4-144*x**3)/exp(x),x)

[Out]

-36*x**4*exp(-x) + 8*x**3 - 3*x**2*log(7) + 15*x

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Giac [A]
time = 0.41, size = 25, normalized size = 0.93 \begin {gather*} -36 \, x^{4} e^{\left (-x\right )} + 8 \, x^{3} - 3 \, x^{2} \log \left (7\right ) + 15 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x*log(7)+24*x^2+15)*exp(x)+36*x^4-144*x^3)/exp(x),x, algorithm="giac")

[Out]

-36*x^4*e^(-x) + 8*x^3 - 3*x^2*log(7) + 15*x

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Mupad [B]
time = 0.08, size = 25, normalized size = 0.93 \begin {gather*} 15\,x-36\,x^4\,{\mathrm {e}}^{-x}-3\,x^2\,\ln \left (7\right )+8\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)*(exp(x)*(24*x^2 - 6*x*log(7) + 15) - 144*x^3 + 36*x^4),x)

[Out]

15*x - 36*x^4*exp(-x) - 3*x^2*log(7) + 8*x^3

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